After staring at this for some time, I believe you meant

$\displaystyle \begin{pmatrix}1 & a & b+c\\ 1 & b & a+c \\ 1 & c & b+a\end{pmatrix}$

Just to clarify, do you need row echelon form, or reduced row echelon form?

Did your professor not explain the "rules" for manipulating matrices to get echelon form? The rules are to use elementary row operations, of which there are three, (1) switch rows, (2) multiply a row by a nonzero constant, (3) add a multiple of one row to another row. These are called row switching, row multiplication, and row addition, respectively, and a more formal treatment is given

here.